Anyway to rig this wheel?
To answer your question more fully, I switched from comments to an answer.

The first thing I did was created and rigged a very rough estimate of your model:



I then added a square-ish wheel underneath it (visible in above picture). I parented this wheel to the armature via Object (Keep Transform), not any of the Armature Deform settings. This means that when you move the armature, the wheel will move with it (assuming the transformation is in object mode). While the wheel moves in conjunction with the rest of the model, you are free to rotate it on the appropriate axis. Here is what I achieved (glitch is in gif alone):